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\author{UID \underline{\hspace{4cm}} \hspace{1cm} NAME \underline{\hspace{4cm}} }
%\author{Class 2019 Math and Applied Math }
\title{Applied Stochastic Processes - Quiz 02}
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%\date{2021 年 2 月 28 日}
\date{April 27, 2021}

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\begin{document}

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\begin{enumerate}

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\item
Components in a machine fail and are replaced according to a Poisson process of rate 3 components a month.
\begin{enumerate}
\item   Find the probability that exactly 3 components fail in the first month and exactly 5 components fail in the next two months.
\item   Find the probability that exactly 3 components fail in the first month given that exactly 8 components fail in the first three months.
\end{enumerate}


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\item
The first Line 9 train leaves the GT Station at 6am. People arrive at the GT Station according to a Poisson process of rate 2 people per minute, starting from 5:30. 
\begin{enumerate}
\item   Determine the mean total number of the people taking the first train at this station.
\item   Determine the mean total waiting time of the people taking the first train.
\end{enumerate}


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\item
Let $X$ and $Y$ be two independent exponential random variables with parameters $\mu>0$ and $\lambda>0$ respectively.
Calculate $P(\min(X,Y)>r)$ for $r\ge 0$. Hence show that $\min(X,Y)$ also has an exponential distribution, whose parameter you should determine.


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\item
Suppose that $\{X(t),t\ge 0\}$ is a Poisson process with rate $\lambda>0$, and the arrival times are $W_1, W_2,\cdots$.
Evaluate the following probabilities in terms of $\lambda$.
\begin{enumerate}
\item   $P[X(3) = 5]$.

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\item   $P[X(3)-X(1) = 4]$.

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\item   $P[X(1)\ge 1 \text{ and } X(3)\le 2]$.

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\item   $P(W_1>1/2)$.

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\item   $P(W_1>1/2 \textrm{ and } W_3\le 3/2)$.

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\item   $P(W_1>1 \textrm{ and } W_2-W_1\le 1/2)$.
\end{enumerate}

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\item
Cars pass a certain street location according to a Poisson process with rate $\lambda=3$ cars per minute. A woman who wants to cross the street at that location waits until she can see that no cars will come by in the next minute.
\begin{enumerate}
\item Find the probability that her waiting time is 0.
\item Find her expected waiting time. Hint: Condition on the time of the first car.
\end{enumerate}


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\end{enumerate}


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